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physics
classical dynamics of particles

System Dynamics 3rd edition William Palm III - Solutions

a. Design a PI and an I controller with internal feedback for the plant Gp(s) = 1/4s. See Figure PI 1.30. We are given that mmax = 6 and rmax = 3. Set (= 1.b. Evaluate the unit-step response of each design.c. Evaluate the unit-ramp response of each design.Figure P11.30
Compare the performance of the critically damped controllers shown in Figure P11.30 with the plant Gp(s) = 1 / I s having the following inputs: a. A unit-ramp disturbance b. A sinusoidal disturbance c. A sinusoidal command input
A certain field-controlled de motor with load has the following parameter values. L = 2 × 10-3 H R = 0.6 Ω KT = 0.04N · m/A C = 0 I = 6 × 10-5 kg · m2 Compute the gains for a state variable feedback controller using P action to control the motor's angular position. The desired dominant time
In Figure P11.33 the input u is an acceleration provided by the control system and applied in the horizontal direction to the lower and of the rod. The horizontal displacement of the lower and is y. The linearized from of Newton's law for small angles givesa. Put this model into state variable form
Figure P11.34 illustrates an active vibration control scheme for a two-mass system. An electro hydraulic actuator between the two masses provides a force that acts on both and is under feedback control. The system model ism1 1 = k1(y - x1) - k2 (x1 - x2) - c(1 - 2) - fm2 2 = k2(x1 - x2) + c(1 - 2)
Figure P11.35 a is the circuit diagram of a speed-control system in which the dc motor voltage va is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not so
The following equations are the model of the roll dynamics of a missile ([Bryson, 1975]). See Figure P11.36.Figure P11.36WhereÎ´ = aileron deflectionb = aileron effectiveness constantu = command signal to the aileron actuator( = roll angle, ( = roll rateUsing the specific value b = 10 s-1 and ( =
Many winding applications in the paper, wire, and plastic industries require a control system to maintain proper tension. Figure P11.37 shows such a system winding paper onto a roll. The paper tension must be held constant to prevent internal stresses that will damage the paper. The pinch rollers
An electro-hydraulic positioning system is shown in Figure P11.38. Use the following values.Ka = 10 V/A Ki = 10–2 in./V K2 = 3 × 105 sec–3 K3 = 20 V/in. z = 0.8
a) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use
Sketch the root locus plot of ms2 + 12s + 10 = 0 for m ≥ 2. What is the smallest possible dominant time constant, and what value of m gives this time constant?
The following table gives the measured open-loop response of a system to a unit-step input. Use the process reaction method to find the controller gains for P. PL and PID control. Time
A liquid in an industrial process must be heated with a heat exchanger through which steam passes. The exit temperature of the liquid is controlled by adjusting the rate of flow of steam through the heat exchanger with the control valve. An open-loop test was performed in which the steam pressure
Use MATLAB to obtain the root locus plot of 5s2 + cs + 45 = 0 for c ≥ 0.
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable k ≥ 0. Use the values m = 4 and c = 8. What is the smallest possible dominant time constant and the associated value of k?
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.43 in terms of the variable c ≥ 0. Use the values m = 4 and k = 64. What is the smallest possible dominant time constant and the associated value of c?
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.45 in terms of the variable k2 ‰¥ 0. Use the values m = 2, c = 8, and k1 = 26. What is the value of k2 required to give ( = 0.707?Figure P11.43Figure P11.45 Figure P11.46
Use MATLAB to obtain the root locus plot of the system shown in Figure P11.46 in terms of the variable c2 ≥ 0. Use the values m = 2, C1 = 8, and k = 26. What is the smallest possible dominant time constant and the associated value of c2?
Use MATLAB to obtain the root locus plot of s3 + 13s2 + 52s + 60 + K = 0 for K ≥ 0. Is it possible for any dominant roots of this equation to have a damping ratio in the range 0.5 ≤ ( ≥ 0.707 and an un-damped natural frequency in the range 3 ≤ (n < 5?
(a) Use MATLAB to obtain the root locus plot of 2s3 + 12s2 + 16s + K = 0 for K ≥ 0. (b) Obtain the value of K required to give a dominant root pair having ( = 0.707. (c) For this value of K. obtain the unit-step response and the maximum overshoot, and evaluate the effects of the secondary root.
Use MATLAB to obtain the root locus of the armature-controlled dc motor model in terms of the damping constant c, and evaluate the effect on the motor time constant. The characteristic equation is LaIs2 + (Ra I + cLa)s + cRa + Kb KT = 0 Use the following parameter values: Kb = KT = 0.1 N · m/A I
In the following equations, identify the root locus plotting parameter K and its range in terms of the parameter p, where p ≥ 0. 6s2 + 8s + 3p = 0 3s2 + (6 + p)s + 5 + 2p = 0 4s3 + 4ps2 + 2s + p = 0
Consider the two-mass model shown in Figure P11.50. Use the following numerical values: m1 = m2 = 1, k1 = 1, k2 = 4, and c2 = 8.a. Use MATLAB to obtain the root locus plot in terms of the parameter c1.b. Use the root locus plot to determine the value of c1 required to give a dominant root pair
Consider the equations3 + 10s2 + 24s + K = 0a. Use MATLAB to obtain the value of K required to give dominant roots with ( = 0.707. Obtain the three roots corresponding to this value of K.b. Use MATLAB to obtain the value of K required to give a dominant time constant of ( = 2/3. Obtain the three
Consider the equations3 + 9s2 + (8 + K)s + 2 K = 0a. Use MATLAB to obtain the value of K required to put the dominant root at the breakaway point. Obtain the three roots corresponding to this value of K.b. Investigate the sensitivity of the dominant root when K varies by ± 10% about the value
Consider the equation s3 + 10s2 + 24s + K = 0 Use the sgrid function to determine if it is possible to obtain a dominant root having a damping ratio in the range 0.5 ≤ ( ≥ 0.707, and an un-damped natural frequency in the range 2 ≤ (n ≤ 3. If so, use MATLAB to obtain the value of K required
In Example 10.7.4 the steady-state error for a unit-ramp disturbance is 1/KI.For the gains computed in that example, this error is 1 /25. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and KD in that example, obtain a root locus plot with KI as the
In Example 10.8.3 the steady-state error for a unit-ramp command is -4/KI. For the gains computed in that example, this error is 1 /1000. We want to see if we can make this error smaller by increasing KI. Using the values given for Kp and Kd in that example, obtain a root locus plot with K1 as the
With the PI gains set to Kp = 6 and K1 = 50 for the plant Gp (s) = 1 / s + 4 The time constant is ( = 0.2 and the damping ratio is ( = 0.707. a. Suppose the actuator saturation limits are ( 5. Construct a Simulink model to simulate this system with a unit-step command. Use it to plot the output
Consider a unity feedback system with the plant Gp(s) and the controller Gc(s). PID control action is applied to the plant Gp(s) = s + 10 / (s + 1) (s + 2) The PID controller has the transfer function Gc(s) = Kp (1 + 1/TIs + TDs Use the values TI = 0.2 and TD = 0.5. Identify the open-loop poles and
With the PI gains set to Kp = 6 and K1 = 50 for the plant Gp (s) = 1 / s + 4 The time constant is ( = 0.2 and the damping ratio is ( = 0.707. Suppose there is a rate limiter of ± 0.1 between the controller and the plant. Construct a Simulink model of the system and use it to determine the effect
A certain dc motor has the following parameter values:L = 2 Ã— 10-3 HR = 0.6 Î©KT = 0.04 N · m/Ac = 0I = 6 Ã— 10-5 kg · m2Figure P11.61Figure P11.61 shows an integral controller using state-variable feedback to control the motor's angular position.a. Compute the gains to give a dominant
Consider the liquid-level controller designed in Example 10.10.1, whose Simulink diagram is shown in Figure 10.10.1. Modify the model to include a Rate Limiter block to limit the rate of q1, in front of the Saturation block. The limits on the rate should be ± 20. Use this model to obtain plots of
In parts (a) through (f), sketch the root locus plot for the given characteristic equation for K ≥ 0.s(s + 5) + K .= 0s(s + 7)(s + 9) + K = 0s2 + 3s + 5 + K(s + 3) = 0s(s + 4) + K(s + 5) = 0s(s2 + 3s + 5) + K = 0s{s + 3)(s + T) + K(s + 4) = 0
PID control action is applied to the plant Gp(s) = s + 10/ (s + 2) (s + 5) The PID controller has the transfer function Gc(s) = Kp (1 + 1/TIs + TDs) Use the values TI = 0.2 and TD = 0.5. Plot the root locus with the proportional gain Kp as the parameter.
Consider the following equation where the parameter p is nonnegative. 4s3 + (25 + 5p)s2 + (16 + 30p)s + 40p = 0 Put the equation in standard root locus form and define a suitable root locus parameter K in terms of the parameter p. Obtain the poles and zeros, and sketch the root locus plot.
Control of the attitude 0 of a missile by controlling the fin angle cp. as shown in Figure P12.1, involves controlling an inherently unstable plant. Consider the specific plant transfer functionGp(s) = ( (s) / ( (s) = 1 / 5s2 - 6Design a PD compensator for this plant. The dominant roots of the
Consider a plant whose open-loop transfer function is G(s) H (s) = 1 / s [(s + 2)2 + 9] The complex poles near the origin give only slightly damped oscillations that are considered undesirable. Insert a gain Kc and a compensator Gc (s)in series to speed up the closed-loop response of the system.
a) The equations of motion of the inverted pendulum model were derived in Example 3.5.6 in Chapter 3. Linearize these equations about ( = 0, assuming that ( is very small, b) Obtain the linearized equations for the following values: M = 10 kg, m = 50 kg, L = 1 m, I = 0, and g = 9.81 m/s2. c) Use
A certain unity feedback system has the following open-loop system transfer function.G(s) = 5K / s3 + 6s2 + 5sObtain the Bode plots and compute the phase and gain margins fora. K = 2b. K =20c. Use the Bode plots to determine the upper limit on K for the system to be stable. Which is the limiting
Figure P12.13 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume thatFigure P12.13The pressure p2 is constant, and consider the specific plant Gp(s) = X (s)/P1 (s) = 1 / 100s2 + s With the following series PD
The height h2 in Figure P12.14 can be controlled by adjusting the flow rate q1. Consider the specific plantGp (s) = H2 (s) / Q1 (s) = 25 / 5s2 + 6s + 5With the following series PI compensator is used to control it.Gc (s) = 7s + 64 / sObtain the Bode plots for this system, and determine the phase
Rolling motion of a ship can be reduced by using feedback control to vary the angle of the stabilizer fins, much like ailerons are used to control aircraft roll. Figure PI2.15 is the block diagram of a roll control system in which the roll angle is measured and used with proportional control
The following transfer functions are the forward transfer function G(s) and the feedback transfer function H(s) for a system whose closed-loop transfer function is G (s) / 1 + G (s) H (s) For each case determine the system type number, the static position and velocity coefficients, and the
Remote control of systems over great distance, such as required with robot space probes, may involve relatively large time delays in sending commands and receiving data from the probe. Consider a specific system using proportional control, where the total dead time is D = D1 + D2 = 100 sec (Figure
Hot-air heating control systems for large buildings may involve significant dead time if there is a large distance between the furnace and the room beingHeated (Figure P12.19). Proportional control applied to the specific plant Gp(s) = 1 / 0.1s + 1 Has the gain KP = 10. The time units are minutes.
Figure PI2.2 shows a pneumatic positioning system, where the displacement x is controlled by varying the pneumatic pressure P1. Assume that the pressure P2 is constant, and consider the specific plantGp (s) = X (s) / P1 (s) = K / s2 + 2sWith K = 4, the damping ratio is ( = 0.5, the natural
The block diagram of a position control system is shown in Figure P12.20. Design a compensator for the particular plantGp (s) = 1 / s(s2 + 3s + 2)So that the static velocity error coefficient will be Cu = 5/sec, the gain margin will be no less than 10 dB, and the phase margin no less than
The speed wi of the load is to be controlled with the torque T acting through a fluid coupling (see Figure P12.5). Design a compensator for the specific plant Gp (s) = Ω2 (s) / T (s) = 4 / s2 + 2s So that the static velocity error coefficient will be Cu = 20/sec, the gain margin will be no less
Design a compensator for the plant Gp (s) = 2 / s2 + 2s So that the static velocity error coefficient will be Cu = 20/sec and the phase margin at least 45o.
Figure P12.2 shows a pneumatic positioning system, where the displacement x is controlled by controlling the applied pneumatic pressure p1. Assume that the pressure p2 is constant, and consider the specific plant Gp (s) = X (s) / P1 (s) = 1 / s2 + s Design a compensator for the plant so that Cu =
The block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plantGp (s) = 1 / s (s + 5) (s + 1)That will give a static velocity error coefficient of Cu = 50/sec and closed loop roots with a damping ratio of ( = 0.5.
The block diagram of a position control system is shown in Figure P12.7. Design a compensator for the particular plant Gp (s) = 1 / s (s2 + 3s + 2) So that the static velocity error coefficient will be Cu = 10/sec, the gain margin will be no less than 10 dB, and the phase margin no less than 50o.
Consider a unity-feedback system having the open-loop transfer functionG(s) = (2n / s(s + 2 ( (n)Derive the following expression for this system's phase margin
Automatic guided vehicles are used in factories and warehouses to transport materials. They require a guide path in the floor and a control system for sensing the guide path and adjusting the steering wheels. Figure P12.28 is a block diagram of such a control system. Obtain the transfer function Gc
With the PI gains set to Kp = 6 and Kt = 50 for the plant Gp (s) = 1 / s + 4 The time constant is ( = 0.2 and the damping ratio is ( = 0.707. a. Compute the gain and phase margins. b. Suppose there is dead time D = 0.1 between the controller and the plant. Compute the gain and phase margins, and
It is desired to control the angular displacement ( of a space vehicle by controlling the applied torque T supplied by thrusters (Figure PI2.3). The plant model is Gp (s) = ( (s) / T (s) = 5/s2 Design a compensator for this plant. The system must have a settling time no greater than 4 sec and a
12.30 In Example 12.1.4 a lag-lead compensator was designed by canceling the plant pole at s = -3 with a compensator zero. Suppose the plant model is slightly inaccurate and the plant pole is really at s = -3.2. Evaluate the resulting unit-step response and unit-ramp response in terms of the
When proportional control is applied to the following plant using a gain of Kp = 1, the closed-loop roots arc satisfactory, but the static velocity error coefficient must be increased to Cu = 5/sec. Gp (s) = 1 / s3 + 3s2 + 2s Design a compensator for this plant that keeps the closed-loop roots near
The speed (1 of the load is to be controlled with the torque Td acting through a fluid coupling (Figure P12.5). Design a compensator for the specific plant Gp(s) = Ω1 (s) / Td (s) = 1 / s2 + s The static velocity error coefficient must be Cu = 10/sec, the dominant roots of the closed-loop system
The block diagram of a speed control system is shown in Figure P12.6. For a particular system with proportional control, G1(s) = KP, the open-loop transfer function isG(s) = Kp / s (s + 2)With Kp = 4, the damping ratio is ( = 0.5, the natural frequency is (n = 2 rad/sec, and Cu = 2/sec. Design a
The block diagram of a position control system is shown in Figure P12.7. For a particular system with proportional control, G1 (s) = Kp, the open-loop transfer function isG (s) = 2.5Kp / s (s + 2) (0.25s + 2)Design a compensator to give s = - 2 ( j 2 (3 and Cu = 80/sec.Figure P12.7
It is desired to control the angular displacement ( of a space vehicle by controlling the applied torque T supplied by thrusters (Figure PI2.8a). The plant model isGp (s) = ( (s) / T (s) = 10 / s2Figure P12.8The feedback sensor that measures the displacement has a time constant of 0.1 sec (Figure
The plant transfer function for the angular displacement ( of an inertia / subjected to a control torque T is (see Figure PI 2.8a) Gp (s) = ( (s) / T (s) = 1 / Is2 Suppose that I = 5 and that the output of the controller is the torque. T. Use the root locus to investigate whether or not a
The 0.5-kg mass shown in Figure P13.1 is attached to the frame with a spring of stiffness k = 500 N/m. Neglect the spring weight and any damping. The frame oscillates vertically with an amplitude of 4 mm at a frequency of 3 Hz. Compute the steady-state amplitude of motion of the mass.Figure P13.1
Alternating-current motors are often designed to run at a constant speed, typically either 1750 or 3500 rpm. One such motor for a power tool weighs 20 lb and is to be mounted at the end of a steel cantilever beam. Static-force calculations and space considerations suggest that a beam 6 in. long, 4
When a certain motor is started, it is noticed that its supporting frame begins to resonate when the motor speed passes through 900 rpm. At the operating speed of 1750 rpm the support oscillates with an amplitude of 8 mm. Determine the amplitude that would result at 1750 rpm if the support were
A 500-lb motor is supported by an elastic pad that deflects 0.25 in. when the motor is placed on it. When the motor operates at 1750 rpm, it oscillates with an amplitude of 0.1 in. Suppose a 1500-lb platform is placed between the motor and the pad. Compute the oscillation amplitude that would
A certain pump weighs 50 lb and has a rotating unbalance. The unbalanced weight is 0.05 lb and has an eccentricity of 0.1 in. The pump rotates at 1000 rpm. Its vibration isolator has a stiffness of k = 500 lb/ft. Compute the force transmitted to the foundation if the isolator's damping ratio is (a)
To calculate the effects of rotating unbalance, we need to know the value of the product mu€, where mu is the unbalanced mass and € is the eccentricity. These two quantities are sometimes difficult to calculate separately, but sometimes an experiment can be performed to estimate the product
A computer disk drive is mounted to the computer's chassis with an isolator consisting of an elastic pad. The disk drive motor weighs 3 kg and runs at 3000 rpm. Calculate the pad stiffness required to provide a 90% reduction in the force transmitted from the motor to the chassis.
Figure P13.16 shows a motor mounted on four springs (the second pair of springs is behind the front pair and is not visible). Each spring has a stiffness k = 2000 N/m. The distance D is 0.2 m. The inertia of the motor is I = 0.2 kg · m2; its mass is m = 25 kg, and its speed is 1750 rpm. Because
A motor mounted on a cantilever beam weighs 20 lb and runs at the constant speed of 3500 rpm. The steel beam is 6 in. long, 4 in. wide, and 3/8 in. thick. The unbalanced part of the motor weighs 1 lb and has an eccentricity of 0.01 ft. The damping in the beam is very slight. Design a vibration
A motor mounted on a beam vibrates too much when it runs at a speed of 6000 rpm. At that speed the measured force produced on the beam is 60 lb. Design a vibration absorber to attach to the beam. Because of space limitations, the absorber's mass cannot have an amplitude of motion greater than 0.08
The supporting table of a radial saw weighs 160 lb. When the saw operates at 200 rpm it transmits a force of 4 lb to the table. Design a vibration absorber to be attached underneath the table. The absorber's mass cannot vibrate with an amplitude greater than 1 in.
A quarter-car representation of a certain car has a stiffness k = 2000 lb/ft, which is the series combination of the tire stiffness and suspension stiffness, and a damping constant of c = 360 lb-sec/ft. The car weighs 2000 lb. Suppose the road profile is given (in feet) by y(t) = 0.03 sin (t, where
A certain machine of mass 8 kg with supports has an experimentally determined natural frequency of 6 Hz. It will be subjected to a rotating unbalance force with an amplitude of 50 N and a frequency of 4 Hz. a. Design a vibration absorber for this machine. The available clearance for the absorber's
The operating speed range of a certain motor is from 1500 to 3000 rpm. The motor and its mount vibrate excessively at 2100 rpm. When a vibration absorber weighing 5 lb and tuned to 2100 rpm was attached to the motor, resonance occurred at 1545 and 2850 rpm. Design a more effective absorber that
Figure P13.22 shows another type of vibration absorber that uses only mass and damping, and not stiffness, to reduce vibration. The main mass is m1 and the absorber's mass is m2. Suppose the applied force f(t) is sinusoidal.Derive the expressions for X1(j()/F(j() and X2(j()/F(j().Use these
Figure P13.23 shows another type of vibration absorber that uses mass, stiffness, and damping to reduce vibration. The damping can be used to reduce the amplitude of motion near resonance. The main mass is m1 and the absorber's mass is m2. Suppose the applied force f(t) is sinusoidal.a. Derive the
Find and interpret the mode ratios for the system shown in Figure P13.24. The masses are m1 = 10 kg and m2 = 30 kg. The spring constants are k11 = 104 N/m and k2 = 2Ã—104 N/m.
Find and interpret the mode ratios for the coupled pendulum system shown in Figure P13.25. Use the values m1 = 1, m2 = 4, L1 = 2, L2 =5, and k = 2.
Find and interpret the mode ratios for the torsional system shown in Figure P13.26. Use the values I1 = 1, I2 = 5, k1 = 1, and k2 = 3.
Find and interpret the mode ratios for the system shown in Figure P13.27.
For the roll-pitch vehicle model described in Example 13.4.2, the suspension stiffnesses are to be changed to k1 = 1.95 × 104 N/m and k2 = 2.3 × 104 N/m. Find the natural frequencies the mode ratios, and the node locations.
A particular road vehicle weighs 4000 lb. Using the quarter-car model, determine a suitable value for the suspension stiffness, assuming that the tire stiffness is 1300 lb/in.
A certain factory contains a heavy rotating machine that causes the factory floor to vibrate. We want to operate another piece of equipment nearby and we measure the amplitude of the floor's motion at that point to be 0.01 m. The mass of the equipment is 1500 kg and its support has a stiffness of k
The vehicle model shown in Figure 13.4.2(a) has the following parameter values: weight = 4800 lb, 1G = 1800 slug-ft2, L1 = 3.5 ft, and L2 = 2.5 ft. Design the front and rear suspension stiffnesses to achieve good ride quality.
A 125-kg machine has a passive isolation system for which c = 5000 N-m/s and k = 7 × 106 N/m. The rotating unbalance force has an amplitude of 100 N with a frequency of 2500 rpm. The resonant frequency of this system is 216 rad/s and is close the frequency of the disturbance. In addition, the
A 20-kg machine has a passive isolation system whose damping ratio is 0.28 and whose un-damped natural frequency is 13.2 rad/s. Assuming that the passive system remains in place, calculate the control gains required to give a damping ratio of ( = 0.707 and a resonant frequency of 141 rad/s.
With the increased availability of powered wheelchairs, improved suspension designs are required for safety and comfort. One chair uses an active suspension like the one shown in Figure P13.33 for each driving wheel.An actuator exerts a force f between the tire and the vehicle. Assuming that the
The following model describes a mass supported by a nonlinear spring. The units are SI, so g = 9.81 m/s2. 5 = 5g - (900y + 1700y3) Find the equilibrium position yr, obtain a linearized model using the equilibrium as the reference operating condition, and compute the oscillation frequency of the
The following is a model of the velocity of an object subjected to cubic damping. m du / dt = - cu3
Find the equilibria of equation (13.6.10) for m = l, c = 12, k1 = 16, and k3 = -4, and use a numerical method to solve and plot the solution for (0) = 0 and four values of y(0): ± 1, ± 1.9, and ± 2.1. Compare the results with the stability properties predicted from the linearized model.
Use a numerical method to compute and plot the free response of equation (13.6.10) for m = 1, c = 0, k1 = 2, and k3 = 0.1, for (0) = 0 and two initial conditions: y(0) = 10 and y(0) = 40. Compare the results with those shown in Figure 13.6.4. How does the initial condition affect the frequency of
Plot the phase plane plots for the following equations and the initial conditions: x(0) = l, (0) = 0.a. + 0.1 + 2x = 0b. + 2+ 2x = 0c. + 4+ 2x = 0
Plot the phase plane plot for the following equation with the initial conditions: y(0) = l, (0) = 0.+ 2+ 2y + 3y3 =0
An electronics module inside an aircraft must be mounted on an elastic pad to protect it from vibration of the airframe. The largest amplitude vibration produced by the airframe's motion has a frequency of 40 Hz. The module weighs 200 N, and its amplitude of motion is limited to 0.003 m to save
Plot the phase plane plot and identify the limit cycle for Van der Pol's equation (13.6.13) with μ = 5 and the initial conditions: y(0) = 1, (0) = 0.
An electronics module used to control a large crane must be isolated from the crane's motion. The module weighs 2 lb. (a) Design an isolator so that no more than 10% of the crane's motion amplitude is transmitted to the module. The crane's vibration frequency is 3000 rpm. (b) What percentage of the

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